# Is this transitive relation or not? [closed]

Is A = {(1,1) (1,2) (2,1)} a transitive relation on {1,2}?

It is confusing. Yes and No both seems to be right. I only need a hint.

• My hint is: First write down the definition of a transitive relation. It is a condition that begins "For all $x$, $y$, and $z$, …". There are only 8 possible choices of $x$, $y$, and $z$. Then check the condition for each of the 8 possible cases. If they are all true, the relation is transitive. – MJD Apr 30 '12 at 13:46
• When I said "write down" I really meant that you should write it, not that you should read it from the book, or imagine it. I hope this was clear. – MJD Apr 30 '12 at 14:02
• en.wikipedia.org/wiki/Transitive_relation – Peđa Terzić Apr 30 '12 at 14:03
• 2 is related to 1, and 1 is related to 2, therefore transitivity would say 2 is related to 2. But it isn't. – Michael Hardy Apr 30 '12 at 15:19
• You might want to explain why you think that both Yes and No are right. – Ben Millwood May 27 '12 at 16:55

Hint: $(2,2)$ is not a member of $A$.

Hint: if it were "no", you would be able to find a pair (a,b), (b,c), but you would be missing the pair (a,c).

• so you are saying that the answer is yes. – Faisal Apr 30 '12 at 13:52
• I'm giving you a hint to try to get you to stop thinking the answer is "Yes and No". Mark Dominus' advice above is good, try it out! – rschwieb Apr 30 '12 at 13:54
• That is one of the 8 things to check. Now do the other 7. – GEdgar Apr 30 '12 at 14:09
• In the present situation, the other 4, really. – Did Apr 30 '12 at 14:17
• You already checked 121. Remain 111, 112, 212 and 211 (making 4 (not 3) more to check). When more experienced, you will see that xxy and xyy cannot be counterexamples to transitivity (can you show why?). Which would leave 121 (already checked) and 212, and... nothing else. – Did Apr 30 '12 at 14:26